Squares & Square Roots (Long Division Method) | Mathematics

Chapter 1 · Number Operations

1.1.1 Squares & Square Roots

Squaring a number is easy — multiply it by itself. Square ROOTING a six-digit number by hand looks impossible, until you see it's just long division wearing a disguise.

🎯 By the end of this you can…

✓Square any number, including negative ones, without a calculator
✓Find the square root of a large perfect square using long division
✓Group digits in pairs correctly before you start dividing

💡 The idea

Squaring is multiplying by itself

$n^{2} = n \times n$ . A negative number squared is always positive, because a negative times a negative is a positive: $(- 3)^{2} = (- 3) \times (- 3) = 9$ , not $- 9$ .

A perfect square is a whole number that is some whole number multiplied by itself — 1, 4, 9, 16, 25, 36… all the way up. Finding its square root is finding which whole number that was.

📋 Square root by long division

1
Split the number into pairs of digits, counting from the right (the units end) — not the left. $203401$ becomes $20 ∣ 34 ∣ 01$ .
2
Find the biggest digit whose square fits in the first pair (or single leading digit). Subtract, then bring down the next pair.
3
Double the root found so far. Find the next digit $d$ so that (double-the-root, with $d$ stuck on the end) $\times d$ still fits under what you brought down.
4
Repeat step 3 until every pair has been used. The digits you wrote down, in order, are the square root.

🧮 √203401 by long division — watch it, one row at a time

⭐ 5 XP

Root so far:

1First pair:

20

. Find the biggest digit

d

so that

d^{2} \leq 20

.4 × 4 = 16Subtract:

20 - 16 = 4

.write 4

💡Why pair from the right? A perfect square's digit-count always lands exactly on a pair boundary counted from the units end — that's what makes the FIRST pair sometimes a single leftover digit, never the last one.

row 1 / 3

✏️ Worked example

Find √105625

1
Pair from the right: $10 ∣ 56 ∣ 25$ .
2
First digit: $3^{2} = 9 \leq 10$ , $4^{2} = 16$ too big, so root starts with 3, remainder $1$ , bring down $56 \to 156$ .
3
Double 3 = 6. $(60 + 2) \times 2 = 124 \leq 156$ , $63 \times 3 = 189$ too big — next digit is 2, remainder $32$ , bring down $25 \to 3225$ .
4
Double 32 = 64. $(640 + 5) \times 5 = 3225$ exactly — last digit 5, remainder $0$ .

✅ Answer: $105625 = 325$

⚠️ The classic trap

Pairing from the left instead of the right is the single most common mistake. $203401$ paired wrong as $2 ∣ 03 ∣ 40 ∣ 1$ gives a completely different (wrong) set of groups. Always start counting pairs from the units digit and work left — if the last group on the left only has one digit, that's fine, leave it alone.

🎮 Find each square root by long division

⭐ 0 XP🔥 streak 0

Find

203401

Find

182329

Find

389376

Find

15129

Find

7056

What is

(- 7)^{2}

0 of 6 finished

🤔 When finding √203401, after pairing the digits (20|34|01), what is the biggest digit d with d² ≤ 20?

🏅

You finished this lesson!

You can now reverse a square by hand — a skill the worksheet uses again the moment a word problem gives you an area and asks for a side.

Keep reading — this lesson marks itself done automatically.